Fox H-function

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

$H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , A_1 ) & ( a_2 , A_2 ) & \ldots & ( a_p , A_p ) \\ ( b_1 , B_1 ) & ( b_2 , B_2 ) & \ldots & ( b_q , B_q ) \end{matrix} \right. \right] = \frac{1}{2\pi i}\int_L \frac {(\prod_{j=1}^m\Gamma(b_j+B_js))(\prod_{j=1}^n\Gamma(1-a_j-A_js))} {(\prod_{j=m+1}^q\Gamma(1-b_j-B_js))(\prod_{j=n+1}^p\Gamma(a_j+A_js))} z^{-s} \, ds$

where L is a certain contour separating the poles of the two factors in the numerator. Another generalization of Fox H-function is given by Innayat Hussain AA (1987). For a further generalization of this function, useful in Physics and Statistics, see Rathie (1997).

The special case for which the Fox H-function reduces to the Meijer G-function is A_j = B_k = C, C > 0 for j = 1...p and k = 1...q (Srivastava 1984, p. 50):

$H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix} ( a_1 , C ) & ( a_2 , C ) & \ldots & ( a_p , C ) \\ ( b_1 , C ) & ( b_2 , C ) & \ldots & ( b_q , C ) \end{matrix} \right. \right] = \frac{1}{C} G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z^{1/C} \right).$

References

• Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society 98: 395–429, doi:10.2307/1993339, ISSN 0002-9947, JSTOR 1993339, MR 0131578

• Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", J. Phys. A: Math. Gen. 20: 4109–4117

• Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", J. Phys. A: Math. Gen. 20: 4119–4128

• Kilbas, Anatoly A. (2004), H-Transforms: Theory and Applications, CRC Press, ISBN 978-0415299169

• Mathai, A. M.; Saxena, Ram Kishore (1978), The H-function with applications in statistics and other disciplines, Halsted Press [John Wiley & Sons], New York-London-Sidney, ISBN 978-0-470-26380-8, MR 513025
• Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), The H-function, Berlin, New York: Springer-Verlag, ISBN 978-1-4419-0915-2, MR 2562766
• Rathie, Arjun K. (1997), "A new generalization of generalized hypergeometric function", Le Matematiche LII: 297–310.
• Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), The H-functions of one and two variables, New Delhi: South Asian Publishers Pvt. Ltd., MR 691138
• Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.